On special ruled surface pairs in fractional calculus

Year 2025, Volume: 74 Issue: 2, 267 - 276, 19.06.2025

Zehra Nur Koçak , Emel Karaca

https://doi.org/10.31801/cfsuasmas.1500845

https://izlik.org/JA53KX49DH

Abstract

In this paper, we extend the theory of special fractional curve pairs (i.e., F-Bertrand, FMannheim, and F-involute-evolute curve pairs) to fractional ruled surfaces with the perspective of fractional calculus. Next, we characterize two fractional ruled surfaces, offset in the senses of F-Bertrand, F-Mannheim, and F-involute-evolute. Moreover, considering the chain rules in fractional calculus, some significant theorems are proved, and the developability conditions are examined by calculating the distribution parameters. Finally, we give examples to verify the results.

Keywords

Fractional derivative and integrals , ruled surface , special surfaces

References

• Abdeljawad, T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, 279 (2015), 57-66. https://doi.org/10.1016/j.cam.2014.10.016.
• Akgül, A., Khoshnawb, S. H. A., Application of fractional derivative on non-linear biochemical reaction models, International Journal of Intelligent Networks, 1 (2020), 52-58. https://doi.org/10.1016/j.ijin.2020.05.001.
• Caputo, M., Mainardi, F., Linear models of dissipation in anelastic solids, La Rivista del Nuovo Cimento, 1 (1971), 161-198. https://doi.org/10.1007/BF02820620.
• Chen, W., Sun, H., Li, X., Fractional Derivative Modeling in Mechanics and Engineering, Springer, Singapore, 2022. https://doi.org/10.1007/978-981-16-8802-7.
• Durmaz, H., Özdemir, Z., Şekerci, Y., Fractional approach to evolution of the magnetic field lines near the magnetic null points, Physica Scripta, 99 (2024). https://doi.org/10.1088/1402-4896/ad1c7e.
• Gözütok, U., C¸ oban, H. A., Sağıroğlu, Y., Frenet frame with respect to conformable derivative, Filomat, 33(6) (2019), 1541-1550. https://doi.org/10.2298/FIL1906541G.
• Has, A., Yılmaz, B., Ayvacı, K. H., $C_{\alpha}-$− ruled surfaces respect to direction curve in fractional differential geometry, Journal of Geometry, 115(11) (2024), 1-18. https://doi.org/10.1007/s00022-023-00710-5.
• Has, A., Yılmaz, B., Special fractional curve pairs with fractional calculus, International Electronic Journal of Geometry, 15(1) (2022), 132-144. https://doi.org/10.36890/IEJG.1010311.
• Has, A., Yılmaz, B., Effect of fractional analysis on some special curves, Turkish Journal of Mathematics, 47(5) (2023), 1423-1436. https://doi.org/10.55730/1300-0098.3438.
• Has, A., Yılmaz, B., Effect of fractional analysis on magnetic curves, Revista mexicana de f´ısica, 68(4) (2022), 1-15. https://doi.org/10.31349/RevMexFis.68.041401.
• Has, A., Yılmaz, B., $C_{\alpha}-$− helices and $C_{\alpha}-$− slant helices in fractional differential geometry, Arabian Journal of Mathematics, 13 (2024), 291-301. https://doi.org/10.1007/s40065-024-00460-5.
• Has, A., Yılmaz, B., $C_{\alpha}-$− curves and their $C_{\alpha}-$− frame in conformable differential geometry, Journal of Universal Mathematics, 7(2) (2024), 99-112. https://doi.org/10.33773/jum.1508243.
• Has, A., Yılmaz, B., Measurement and calculation on conformable surfaces, Mediterranean Journal of Mathematics, 20 (2023), 1-18. https://doi.org/10.1007/s00009-023-02471-6.
• Jumarie, G., On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling, Central European Journal of Physics, 11(6) (2013), 617-633. https://doi.org/10.2478/s11534-013-0256-7.
• Kasap, E., Yüce, S., Kuruoğlu, N., The involute-evolute offsets of ruled surfaces, Iranian Journal of Science and Tecnology, Transaction A, 33(2) (2009), 195-201. 10.22099/ijsts.2009.2215.
• Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70. http://dx.doi.org/10.1016/j.cam.2014.01.002.
• Magin, R. L., Fractional calculus in bioengineering, Crit Rev Biomed Eng., 32(1) (2004), 1-104. https://doi.org/10. 1615/critrevbiomedeng.v32.i1.10. PMID: 15248549.
• Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. https://api.semanticscholar.org/CorpusID:117250850.
• Orbay, K., Kasap, E., Aydemir, İ., Mannheim offsets of ruled surfaces, Mathematical Problems in Enginnering, (2009), 1-9. https://doi.org/10.1155/2009/160917.
• Podlubny, I., Fractional Differential Equations, Academic Pres, New York, 1999. https://doi.org/10.1007/978-3-030-76043-4.
• Ravani, B., Ku, T. S., Bertrand offsets of ruled and developable surfaces, Computer-Aided Design, 23(2) (1991), 145-152. https://doi.org/10.1016/0010-4485(91)90005-H.
• Syouri, S. T. R., Sulaiman, I. M., Mamat, M., Abas, S. S., Ahmad, M. Z., Conformable fractional derivative and its application to partial fractional derivatives, J. Math. Comput. Sci., 11(3) (2021), 3027-3036. https://doi.org/10. 28919/jmcs/5655.
• Trasov, V. E., On chain rule for fractional derivatives, Communications in Nonlinear Science and Numerical Simulations, 30(1) (2016), 1-4. https://doi.org/10.1016/j.cnsns.2015.06.007.
• Uchaikin, V. V., Fractional Derivatives for Physicists and Engineers, Springer, Berlin, Heidelberg, 2013. https://doi. org/10.1007/978-3-642-33911-0.
• Yajima, T., Oiwa, S., Yamasaki, K., Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas, Fractional Calculus and Applied Analysis, 21 (2018), 1493-1505. https://doi.org/10.1515/fca-2018-0078.
• Yılmaz, B., A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus, Optik, 247 (2021), 168026. https://doi.org/10.1016/j.ijleo.2021.168026.
• Yılmaz, B., Has, A., Obtaining fractional electromagnetic curves in optical fiber using fractional alternative moving frame, Optik, 260 (2022), 169067. https://doi.org/10.1016/j.ijleo.2022.169067.